The problem of modeling acoustic waves scattered by an object with
Neumann boundary condition is considered. The boundary condition is
taken into account by means of the fictitious domain method, yielding
a first order in time mixed variational formulation for the
problem. The resulting system is discretized
with two families of mixed finite elements that are compatible with
mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used.
We, therefore, introduce the second family of mixed finite elements for which a
theoretical convergence analysis is presented and error estimates are
obtained. A numerical study of the convergence is also considered for
a particular object geometry which shows that our theoretical
error estimates are optimal.